We show that from dimension six onwards (but not in lower dimensions), there are in each dimension flat manifolds with first Betti number equal to zero admitting Anosov diffeomorphisms. On the other hand, it is known that no flat manifolds with first Betti number equal to one support Anosov diffeomorphisms. For each integer k > 1 however, we prove that there is an n-dimensional flat manifold M with first Betti number equal to k carrying an Anosov diffeomorphism if and only if M is a k-torus or n is greater than or equal to k + 2.